Intersection of Random Sets 3

We prove compact-randomness of the intersection of a measurable familyof compact-random sets. This generalizes results on the intersection of countable families of random sets. The main tools are given by the theory of Souslin spaces and the idea of countable separation. An example of a random set is given by the energy spectrum of an electron in a disordered solid, or more generally by the spectrum of a closed linear random operator. If the solid is ergodic the spectrum equals a xed set. This self-averaging result fails if the electron in addition is innuenced by an electric eld. Our idea is to single out the part of the spectrum which depends on the bulk properties of the solid. The intersection result makes it possible to prove self-averaging for this part of the spectrum. This work indicates the importance of the link between the theory of random sets and the theory of linear random operators.